72.11.8 problem 10

Internal problem ID [14762]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.4 page 310
Problem number : 10
Date solved : Thursday, March 13, 2025 at 04:18:28 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-4 x \left (t \right )+6 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 32

ode:=[diff(x(t),t) = 2*x(t)+2*y(t), diff(y(t),t) = -4*x(t)+6*y(t)]; 
ic:=x(0) = 1y(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} \cos \left (2 t \right ) \\ y &= {\mathrm e}^{4 t} \left (\cos \left (2 t \right )-\sin \left (2 t \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 35

ode={D[x[t],t]==2*x[t]+2*y[t],D[y[t],t]==-4*x[t]+6*y[t]}; 
ic={x[0]==1,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{4 t} \cos (2 t) \\ y(t)\to e^{4 t} (\cos (2 t)-\sin (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(4*x(t) - 6*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{4 t} \sin {\left (2 t \right )} + \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{4 t} \cos {\left (2 t \right )}, \ y{\left (t \right )} = - C_{1} e^{4 t} \sin {\left (2 t \right )} + C_{2} e^{4 t} \cos {\left (2 t \right )}\right ] \]